Strain Rate Dependent Constitutive Modelling of 3D Printed Polymers

EPJ Web of Conferences 250, 02029 (2021) https://doi.org/10.1051/epjconf/202125002029
DYMAT 2021

 Strain Rate Dependent Constitutive Modelling of
 3D Printed Polymers
 Maria Lißner1*, Daniel Thomson1, Nik Petrinic1, and Jeroen Bergmann1
 1 Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ

 Abstract. Experimental results from 3D printed TPC (thermoplastic
 copolyester) compression specimens were used to develop a combined
 experimental-numerical framework to support the design of e.g. 3D printed
 mouthguards. First, a commercially available material model capable of
 representing the strain-rate dependent behaviour of materials undergoing
 large deformations is identified. Second, experimental results from solid 3D
 printed compression specimens are used to calibrate the identified material
 models. Third, 3D printed compression specimens with two different cavity
 geometries are used to assess the ability of the material model to accurately
 reproduce the experimental observations. The numerical investigation
 indicates a good representation of the strain rate dependent experimental
 results of 3D printed specimens.

 1 Introduction
 Polymeric soft materials are widely used for protection in many sports, e.g. as mouthguards
 in boxing or as liners in hockey helmets. This is due to their advantageous material properties
 such as high energy dissipation, low density and low weight. Nevertheless, providing
 protection from potentially life-threatening impacts in contact sports (e.g., boxing, rugby) or
 sports with hard ball contact (e.g., field hockey, cricket) can be extremely challenging and
 requires continuous innovation and improvement of existing designs. In addition, many other
 types of potentially debilitating head injuries, such as sport related concussion, intra cranial
 bleeding (ICB), and traumatic brain injury (TBI) cannot be fully prevented even if protective
 equipment such as helmets or mouthguards are already in place [1]. Some of these can be
 attributed to the inability of sports equipment to proactively inform the athlete about their
 health management.

 To address this challenge, wearable sensor technologies (i.e. GPS, or heart monitoring
 systems) are increasingly being considered to monitor the health performance of athletes
 during competition [2,3]. Nevertheless, their positioning is of high importance in order to
 measure the performance of interest as accurately as possible while remaining mechanically
 sound. The integration of sensors into protective equipment provides a suitable approach to
 meet these requirements [4]. Additive manufacturing technology is a potential tool that can
 help achieve this: Cavities to embed the sensors can be printed precisely while minimising

 * Corresponding author: maria.lissner@eng.ox.ac.uk

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons
Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).

EPJ Web of Conferences 250, 02029 (2021) https://doi.org/10.1051/epjconf/202125002029
DYMAT 2021

 excess material. However, the 3D printed safety devices still need to perform reliably under
 demanding mechanical loads - the impact protection of these protective systems must not be
 compromised. This demands a fundamental understanding of the mechanical performance of
 the devices, which must be resistant to impact scenarios and at the same time be able to
 accommodate sensor systems.

 The aim of this work is to develop a numerical-experimental framework that enables the
 reliable design of smart sports protection devices (such as mouthguards). First, a
 commercially available material model is identified that is able to represent the
 experimentally observed strain rate dependent response of the materials. Second,
 experimental results of 3D printed solid compression specimens are used to calibrate the
 identified material model. Third, 3D printed compression specimens with two different cavity
 geometries are used to evaluate the ability of the calibrated material model to accurately
 represent the experimental observations. Finally, the results of the numerical investigations
 are discussed in terms of their potential for reliable predictive design modelling of larger
 scale applications.

 2 Material and Methods
 A thermoplastic material was tested at different strain rates using different topologies. The
 compression tests were conducted using the Split Hopkinson Pressure Bar and Instron
 machine setup. The numerical investigation is based on this experimental campaign using the
 finite element software ABAQUS/Explicit. Experimental results from solid compression
 specimens are used to calibrate material models suitable to represent both the observed large
 strain and the non-linear strain rate dependence. The calibrated material model is then
 verified by modelling the other two configurations (cubic and spherical topologies).

 2.1 Material and specimen designs

 The investigated 3D printable material Arnitel® ID 2045 Natural (DSM, The Netherlands)
 is a TPC (Thermoplastic Copolyester) that is highly flexible and has excellent energy return
 suitable for sports protective equipment.

 The compression specimens have a diameter of 12 mm and a thickness of 4 mm which is
 similar to the thickness of mouthguards. The air cells for the two different topology structures
 are 3x2x2mm and Ø 2mm for the cubic and spherical topology, respectively. The wall
 thickness is 1mm in both cases. Figure 1(a) illustrates the different specimen configurations.

 Figure 1. Graphical illustration of a) the specimen geometries with different topologies and b) the
 model setup.

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EPJ Web of Conferences 250, 02029 (2021) https://doi.org/10.1051/epjconf/202125002029
DYMAT 2021

 2.2 Material model

 2.2.1 Parallel Rheological Framework (PRF)
 The non-linear strain rate behaviour of the investigated material can be described with the
 Parallel Rheological Framework (PRF) introduced by Hurtado et al. [5,6]. This framework
 allows the parallel superposition of viscoelastic and elastic networks. The deformation
 gradient of each network i for finite deformation in elastic materials is assumed as follows
 [5,6]
 = 
 = 1… 
 = 0 (1)

 where represents the elastic part of the deformation gradient and 
 represents the creep
 part of the deformation gradient in the viscoelastic networks. 0 is the elastic component of
 the equilibrium network. Isotropic behaviour and incompressibility are assumed [5,6].

 2.2.2 Elastic Behaviour
 The elastic response of the networks is described with a hyperelastic strain energy potential
 using the principle of virtual work. It is assumed that all networks are described by the same
 strain energy potential formulation combined with the assumption that the total strain energy
 UT is equal to the weighted sum of the strain energies of all the networks as
 = ∑ 
 =0 ( ) (2)
 where si is the stiffness ratio of the ith network, representing the volume fraction of the
 network and = ( ) is the elastic right Cauchy-Green tensor in each network. The
 stiffness ratios as material parameters are required to satisfy the relation
 ∑ 
 =0 = 1 . (3)
 The hyperelastic material behaviour is modelled with the Neo Hooke model which is related
 solely to the first Cauchy strain tensor invariant I1 as follows
 = 1 ( 1 − 3) (4)
 where C1 is the material parameter and the first invariant I1 is represented by the primary
 elongations i (i=1,2,3) as
 1 = 21 + 22 + 23 . (5)

 The principal stress  is calculated using the following relationship between the strain energy
 and the elongation
 
 = − , = 1,2,3 (6)
 
 where p is the hydrostatic pressure and obtained through the kinetic boundary conditions,
 which are for the considered uniaxial case as follows
 1 = , 2 = 3 = 0
 1 = , 2 = 3 = −1/2 . (7)

 The hydrostatic pressure p can then be calculated by substituting 2 = 0 into Eq. (6) alongside
 with Eq. (4), which results in

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EPJ Web of Conferences 250, 02029 (2021) https://doi.org/10.1051/epjconf/202125002029
DYMAT 2021

 1
 = 2 1 . (8)
 2

 Substituting Eq. (4) and (8) into Eq. (6) leads to the calculation of 1 as
 1 1
 1 = ( 1 − 2 ) 1 . (9)
 1 2

 With the relationships for the uniaxial case in Eq. (7) the calculation of the uniaxial
 hyperelastic principal stress is as follows
 = 2 1 ( − −2 ) . (10)

 2.2.3 Viscous Behaviour

 The viscoelastic behaviour of the material is defined by the strain hardening power law
 described in [6,7] and supported in ABAQUS following
 1
 ̇ = { [( + 1) ] } +1 (11)

 where A, m and n are material parameters.

 2.2.4 Parameter Identification
 Parameter identification was performed using the nonlinear least square algorithm lsqnonlin
 in MATLAB. First, the hyperelastic material model was calibrated using the results for the
 low-rate solid compression specimens. Second, the strain rate dependent material parameters
 were determined using the experimental results of the solid compression specimens for all
 three loading rates simultaneously. To use the strain hardening potential law in ABAQUS,
 the parameters m and n are defined to be in the ranges [-1,0] and [0,10], respectively. Finally,
 a most adequate set of material parameters was determined that best fits the strain rate
 dependent experimental results.

 2.2.5 Model setup

 The identified material parameters are used to numerically model the strain rate-dependent
 hyperelastic material behaviour of 3D-printed compression specimens. For this purpose, the
 geometries for the three different specimen configurations were modelled in the finite
 element (FE) software ABAQUS/Explicit. Similar to the experiments, the specimens were
 placed between two bars, which were modelled as rigid bodies due to their high stiffness
 compared to the stiffness of the TPC. The contact definition between the specimens and the
 bars is defined as frictionless. The lower rigid bar is fixed (BC 2), while the upper bar is
 loaded with the equivalent displacement curves of the experiments for up to 45% true strain
 (BC 1). Figure 1(b) represents the set-up graphically.

 Six-node tetrahedral elements are used with a mesh size of 0.25mm. The constitutive
 parameters describing the hyperelastic and strain rate dependent material behaviour are
 implemented into the FE for the two networks – hyperelastic (Neo Hooke model) and
 viscoelastic (Strain hardening power law). The true stress-true strain response is obtained at
 the applied load position (BC 1).

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EPJ Web of Conferences 250, 02029 (2021) https://doi.org/10.1051/epjconf/202125002029
DYMAT 2021

 3 Results and Discussion
 The suitability of the identified parameters is evaluated by numerical modelling of the solid
 3D printed compression specimens. The experimental results of 3D printed specimens with
 different cellular structures are used to verify the calibrated material model and to assess the
 suitability of the performed modelling strategy for 3D printed thermoelastic materials.

 3.1 Strain-rate dependent material parameter identification
 Table 1 presents the obtained parameters for the Neo Hooke material and the strain hardening
 power law. Figure 2(a) shows the numerical prediction of the experimentally obtained strain
 rate dependent material behaviour of solid compression specimens. The root mean square
 errors (RMSE) between the experimental (averaged) and numerical representation for the
 different strain rates are highlighted in Figure 2(a). This demonstrates that the Neo Hooke
 model is a simple and suitable tool in combination with the strain hardening power law for
 modelling the strain rate dependent behaviour of the solid TPC material. Nevertheless, due
 to its nature, the stress-strain relationship is represented as mostly linear, which can have a
 negative impact on the correct prediction of the behaviour, depending on the application.

 Table 1. Identified parameters for numerical simulation

 Neo Hooke model Strain hardening power law
 C D A n M sratio
 2.1636 0.0000 0.4245 0.9964 -0.5 0.9

 Figure 2. Strain-rate dependent true stress - true strain experimental and numerical results for
 compression specimens with a) solid, b) spherical air cells and c) cubic air cells.

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EPJ Web of Conferences 250, 02029 (2021) https://doi.org/10.1051/epjconf/202125002029
DYMAT 2021

 3.2 Model verification

 Figure 2(b) and (c) show the numerical representation of the experimental results with the
 errors for the cubic and spherical cell structures, respectively. The experimental results for
 the cubic air cells are slightly underrepresented by the model when compared to the results
 of the spherical air cells. This difference might be two folded: on one hand, the 3D printing
 process might have introduced inhomogeneities and on the other hand, the cubic geometry
 resulted in sharp edges which are most likely the source of stress concentrations. Both
 reasons, might result in this larger discrepancy of the numerical representation.

 4 Conclusion
 The main objective of this study was to demonstrate the suitability of calibrating the strain
 rate dependency of a commercially available hyperelastic material model using 3D printed
 solid specimens experiments to be used to predict the material behaviour of 3D printed
 structures with different topologies. The following conclusions can be drawn from this
 research:

 1. The Neo-Hooke model and strain hardening power law incorporated in ABAQUS
 were successfully calibrated using strain rate dependent experimental results of 3D
 printed solid compression specimens.
 2. The calibrated strain rate dependent constitutive relationship using the PRF was
 successfully verified by predicting the experimental response of strain-rate
 dependent experiments of 3D-printed printed specimens with different cellular
 structures (cubic and spherical).
 3. The presented modelling strategy provides a useful tool when predicting the
 mechanical performance of different 3D printed topologies during the design
 process of larger applications, such as mouthguards.

 References
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